Vanishing singularity in hard impacting systems

Soumya Kundu,Damian Giaouris,Soumitro Banerjee

doi:10.3934/dcdsb.2011.16.319

Abstract

It is known that the Jacobian of the discrete-time map of an impactoscillator in the neighborhood of a grazing orbit depends on thesquare-root of the distance the mass would have gone beyond theposition of the wall if the wall were not there. This results in aninfinite stretching of the phase space, known as the square-rootsingularity. In this paper we look closer into the Jacobian matrix andfind out the behavior of its two parameters---the trace and thedeterminant, across the grazing event. We show that the determinantof the matrix remains invariant in the neighborhood of a grazingorbit, and that the singularity appears only in the trace of thematrix. Investigating the character of the trace, we show that thesingularity disappears if the damped frequency of the oscillator is anintegral multiple of half of the forcing frequency.

Highlights

This paper concerns the dynamics of such mechanical systems in which the elements of the system may undergo impacts with each other
In this work we show that the determinant remains invariant in the immediate neighborhood of the grazing orbit
In this paper we have analyzed the character of the map function of an impact oscillator, as the variation of a parameter drives the system from a non-impacting orbit to an impacting orbit

Summary

Introduction

This paper concerns the dynamics of such mechanical systems in which the elements of the system may undergo impacts with each other. Since at grazing the normal velocity with respect to the boundary becomes zero (v(x) = 0), the reset map can be formulated as. The variation of the parameter μ from a negative to a positive value causes the border collision bifurcation, and the outcome of the bifurcation depends on the trace and the determinant at the two sides of the border In tune with this logic, we explore how the trace and the determinant of the Jacobian matrix change across the grazing condition.

Thus we have

Forcing function amplitude

Let a number m be defined as m

Position of the mass

Conclusions